We discuss the conceptual problem of identifying the natural notions of computability at higher types (over the natural numbers). We argue for an eclectic approach, in which one considers a wide range of possible approaches to defining higher type computability and then looks for regularities. As a first step in this programme, we give an extended survey of the different strands of research on higher type computability to date, bringing together material from recursion theory, constructive logic and computer science. The paper thus serves as a reasonably complete overview of the literature on higher type computability. Two sequel papers will be devoted to developing a more systematic account of the material reviewed here.

Notions of computability at higher types I

Definability and Full Abstraction

We show that the higher-order matching problem is decidable using a game-theoretic argument.

Decidability of higher-order matching

We review some of the history of the computability theory of functionals of higher types, and we will demonstrate how contributions from logic and theoretical computer science have shaped this still active subject.

Computing with functionals—computability theory or computer science?

We present a collection of formalized results about finite nested multisets, developed using the Isabelle/HOL proof assistant. The nested multiset order is a generalization of the multiset order that can be used to prove termination of processes. Hereditary multisets, a variant of nested multisets, offer a convenient representation of ordinals below 0. In Isabelle/HOL, both nested and hereditary multisets can be comfortably defined as inductive datatypes. Our formal library also provides, somewhat nonstandardly, multisets with negative multiplicities and syntactic or-dinals with negative coefficients. We present applications of the library to formalizations of Goodstein's theorem and the decidability of unary PCF (programming computable functions). 1 Introduction In their seminal article on proving termination using multisets [15], Dershowitz and Manna introduced two orders of increasing strength. The multiset order lifts a base partial order on a set A to finite multisets over A. It forms the basis of the multiset path order, which has many applications in term rewriting [41] and automatic theorem proving [1]. The nested multiset order is a generalization of the multiset order that operates on multisets that can be nested in arbitrary ways. Nesting can increase the order's strength: If (A, <) has ordinal type α < 0 , the associated multiset order has ordinal type ω α , whereas the nested order has ordinal type 0 = ω ω ω. .. . In this paper, we present formal proofs of the main properties of the nested multiset order that are useful in applications: preservation of well-foundedness and preservation of totality (linearity). The proofs are developed in the Isabelle/HOL proof assistant [27]. To our knowledge, this is the first development of its kind in any proof assistant. Our starting point is the following inductive datatype of nested finite multisets over a type a (Section 4): datatype a nmultiset = Elem a | MSet ((a nmultiset) multiset) The above Isabelle/HOL command introduces a (unary postfix) type constructor, nmultiset, equipped with two constructors, Elem : a → a nmultiset and MSet : (a nmultiset)multiset → a nmultiset, where a is a type variable and multiset is the type constructor of (finite) multisets.

https://hal.inria.fr/hal-01599176/document

Nested Multisets, Hereditary Multisets, and Syntactic Ordinals in Isabelle/HOL

Unary PCF is decidable

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Unary PCF is decidable